Waring's Problem and the Goldbach Conjecture

We look here at some of the results about Waring's Problem and the Goldbach Conjecture which have been proved since Hardy gave his inaugural lecture at the University of Oxford in 1920.

1. Waring's Problem g(k).

The number g(k) is the least number such that every number is the sum of g(k) or less k-th powers.

In his 1920 inaugural lecture, Hardy knew that g(1) = 1, g(2) = 4 and g(3) = 9. He did not have an exact value for g(k) for k ≥ 4 but he gives bounds. The following has been proved since 1920:

g(4) = 19 was proved in 1986 by Ramachandran Balasubramanian, Jean-Marc Deshouillers, and François Dress in two papers.g(5) = 37 was proved in 1964 by Chen Jingrun.g(6) = 73 was proved in 1940 by S S Pillai.

It is known that g(k) = 2k + [(3/2)k] - 2 for all k ≤ 471,600,000 where [x] is the largest integer less than x. This was proved by J M Kubina and M C Wunderlich, in their paper "Extending Waring's conjecture to 471,600,000" in Math. Comp. 55 (1990), 815-820.

2. Waring's Problem G(k).

The number G(k) is the least number such that for every integer from a certain point onwards is the sum of G(k)or less k-th powers.

Although much progress has been made in determining g(k), there has been much less progress in determining G(k). In his 1920 inaugural lecture, Hardy knew that G(1) = 1, G(2) = 4 and 4 ≤ G(3) ≤ 8. Hardy also knew that 16 ≤ G(4) ≤ 21. The following has been proved since 1920:

G(3) ≤ 7 was proved by Y V Linnik. The result was announced in 1942 in his paper "On the representation of large numbers as sums of seven cubes" in Dokl. Akad. Nauk SSSR35 (1942), 162. A proof is given in Linnik's paper "On the representation of large numbers as sums of seven cubes" in Mat. Sb.12 (1943), 218-224.

G(4) = 16 was proved by Harold Davenport in 1939 in his paper "On Waring's problem for fourth powers" in Ann. of Math.40 (1939), 731-747.

For G(k), 5 ≤ k ≤ 20, we have the following results which, as of January 2017, we believe are the best obtained so far:

k

G(k)

Proved by

Journal

Year

5

≤ 17

Vaughan & Wooley

Acta Math.

1995

6

≤ 24

Vaughan & Wooley

Duke Math. J.

1994

7

≤ 33

Vaughan & Wooley

Acta Math.

1995

8

≤ 42

Vaughan & Wooley

Phil. Trans. Roy. Soc.

1993

9

≤ 50

Vaughan & Wooley

Acta Arith.

2000

10

≤ 59

Vaughan & Wooley

Acta Arith.

2000

11

≤ 67

Vaughan & Wooley

Acta Arith.

2000

12

≤ 76

Vaughan & Wooley

Acta Arith.

2000

13

≤ 84

Vaughan & Wooley

Acta Arith.

2000

14

≤ 92

Vaughan & Wooley

Acta Arith.

2000

15

≤ 100

Vaughan & Wooley

Acta Arith.

2000

16

≤ 109

Vaughan & Wooley

Acta Arith.

2000

17

≤ 117

Vaughan & Wooley

Acta Arith.

2000

18

≤ 125

Vaughan & Wooley

Acta Arith.

2000

19

≤ 134

Vaughan & Wooley

Acta Arith.

2000

20

≤ 142

Vaughan & Wooley

Acta Arith.

2000

To illustrate the progress towards these "up-to-date" results, we give an indication of how the bounds for G(9) have been improved since Hardy gave his 1920 lecture:

≤

Proved by

Journal

Year

949

G H Hardy & J E Littlewood

Math. Z.

1922

824

R D James

Proc. London Math. Soc.

1934

190

H Heilbronn

Acta Arith.

1936

101

T Estermann

Acta Arith.

1937

99

V Narasimhamurti

J. Indian Math. Soc.

1941

96

R J Cook

Bull. London Math. Soc.

1973

91

R C Vaughan

Acta Arith.

1977

90

K Thanigasalam

Acta Arith.

1980

88

K Thanigasalam

Acta Arith.

1982

87

K Thanigasalam

Acta Arith.

1985

82

R C Vaughan

J. London Math. Soc.

1986

75

R C Vaughan

Acta Math.

1989

55

T D Wooley

Ann. of Math.

1992

51

R C Vaughan & T D Wooley

Acta Math.

1995

50

R C Vaughan & T D Wooley

Acta Arith.

2000

It has been shown that the following lower bounds hold

k

G(k)

5

≥ 6

6

≥ 9

7

≥ 8

8

≥ 32

9

≥ 13

10

≥ 12

11

≥ 12

12

≥ 16

13

≥ 14

14

≥ 15

15

≥ 16

16

≥ 64

17

≥ 18

18

≥ 27

19

≥ 20

20

≥ 25

It has been conjectured that these lower bounds are the correct values for G(k).

3. Goldbach Conjecture.

Hardy states the Goldbach Conjecture in his 1920 inaugural lecture as:

Every even number greater than 2 is the sum of two odd primes.

This is sometimes today called the strong Goldbach Conjecture.

The weak Goldbach Conjecture is:

Every odd number greater than 7 is the sum of three odd primes.

In 2013, Harald Helfgott proved Goldbach's weak conjecture; previous results had already shown it to be true for all odd numbers greater than about 2 × 101346.

The strong Goldbach conjecture has been shown to hold for all n up to 4 × 1018. The following table shows the progress towards this:

105

N Pipping

1938

108

M L Stein & P R Stein

1965

2 × 1010

A Granville, J van der Lune & H J J te Riele

1989

4 × 1011

M K Sinisalo

1993

1014

J M Deshouillers, H J J te Riele & Y Saouter

1998

4 × 1014

J Richstein

2001

2 × 1016

T Oliveira e Silva

2003

6 × 1016

T Oliveira e Silva

2003

2 × 1017

T Oliveira e Silva

2005

3 × 1017

T Oliveira e Silva

2005

12 × 1017

T Oliveira e Silva

2008

4 × 1018

T Oliveira e Silva

2012

JOC/EFR January 2017

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Extras/Waring_January 2017.html